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Algebra and Equations
This page describes the basics of algebra and solving simple equations. Advanced readers should feel free to skip this section.Algebra
Take any mathematical expression (an expression is just a string of numbers and mathematical symbols, like "1 + 1" or "4 × (5+6) = 44" ). Sometimes not all of an expression is known, or for whatever reason it's just not practical to write it all down. In this case, it's often convenient to replace part of a mathematical expression with a symbol, such as a letter of the alphabet. Lowercase letters for example, are often used to stand in place of missing numbers, and are appropriately called pronumerals.For example, many equations; such as those found in the back of many bulky physics textbooks, depend on numbers which cannot be written down because the numbers themselves vary depending on the situation in which the equation is used. So, for example, elapsed time in seconds might be represented in an equation as the letter t. That way, a physicist doing an experiment simply replaces the letter t with the number of seconds elapsed; then uses the equation to get the correct answer. Quantities that vary in this way are called variables.
Alternatively, sometimes values that never vary (called constants) are written as pronumerals. For example, the speed of light is 299792458 metres per second precisely; but it's often just written as c. Why? because if you're working with a complicated equation that's half a page long, it's a lot easier, clearer and safer to write c instead of 299792458 m/s each time you refer to the speed of light! It also saves formulating a new equation from scratch if you later decide to use the speed of light in miles per hour instead!
The number that a variable or constant is equal to - the number it represents at a particular moment; is called its value. So, if you say that the variable x has a value of 5; that simply means that it is equal to 5.
Trivia: The word algebra actually comes from the Arabic al jabr, meaning "reunion."
Equations
An equation is any expression that has an "=" (pronounced equals) sign in the middle. For example:- 1 + 1 = 2
- 2 = 1 + 1
- 36 = 36
- 53 + 10 = 10 + 53
- 53 + 10 = 73 − 10
- 2 × t + 31 = 13
- E = mc2
(The expression on the Left Hand Side of the equals sign is often referred to as the LHS, for short. Likewise, RHS stands for Right Hand Side.)
Solving Equations
Take the equation 2 × t + 31 = 13 above. Since we know that the entire LHS is equal to 13, we can work out what t must be equal to for the equation to make sense. Mathematicians call this "solving for t," in order to save on ink and grammar lessons.| Let's begin with this example: | LHS | RHS | |
| We start with the original equation: | 2 × t + 31 | = | 13 |
|
Step 1: Since the LHS and RHS are actually the same number, let's make a new equation, in which the LHS and RHS are both equal to this number minus 31... |
(2 × t + 31) − 31 | = | (13) − 31 |
|
...we can then simplify the LHS and RHS to get: |
2 × t | = | -18 |
|
oh, and by the way, 2 × t can be abbreviated to 2t. Generally,
with
algebraic symbols, if you write a symbol right next to a number or
another symbol, without specifying +, ×, −,
÷,
etc.
then
multiplication is assumed.
|
2t | = | -18 |
|
Step 2: Great, so we now know what 2 times t is. But we just want t itself, in other words, 1 times t. So, since we know that 2t is equal to -18, therefore 2t divided by 2, must be equal to -18 divided by 2. |
2t / 2 | = | -18 / 2 |
|
of course, 2t / 2 (which equals
t ×
2/2) is just equal to t.
Likewise, any
number divided by itself equals 1.
Hence our answer: |
t | = | -9 |
You can solve just about any equation in this way. You can apply any operation to the whole LHS, and as long as you do exactly the same thing to the whole RHS at the same time; you'll never break the equation; since both sides will still be equal to the same number, albeit a different number to what you started with.
Notice how, in solving the above, we started with t buried in the LHS. The LHS didn't just contain t. It contained a t that had been multiplied by 2 and then added to 31. We then proceeded to un-bury t from the LHS, by performing, in reverse order, the opposite operations to those that had already been done to t in the LHS. So we first subtracted 31 from both the LHS and RHS; then we divided the LHS and RHS by 2. In other words, we undid the operations that were burying t, essentially moving the obstruction from the LHS to the RHS.
This was a simple example, but solving equations is a lot like playing Solitaire, where you can shift unwanted cards around, but can't totally get them out of the way unless you match them up right. There are always numbers or pronumerals in the way, but removing an obstruction from one side of the equation causes an equivalent obstruction to appear on the other side. You can't just get rid of them outright. There are plenty of possible moves, but the really useful ones are those that simplify the whole equation. (In the above example, each step produced a simpler equation than the one that preceded it.)
There's one exception to this though. You can always absolutely solve any equation by multilying both the LHS and RHS by zero. But this is completely useless - everyone already knows that zero equals zero! This is called the trivial solution.
Note that if you have a single equation containing two variables, you can't solve for both of them. You need to find a second equation, solve for one of the variables, then replace that variable with its solution in the first equation. Generally, for each unknown variable to be solved, you need an additional, independent equation.
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